{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Coding Back-Substitution"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Here's an upper-triangular matrix $A$ and two vectors $x$ and $b$ so that $Ax=b$.\n",
        "\n",
        "See if you can find $x$ by computation."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[ 0.53359808 -1.66129881  0.31267643 -0.07384466  1.20957795]\n",
            " [-0.         -1.1204435  -1.5348203   1.38270361 -0.34971611]\n",
            " [ 0.          0.         -0.47187693  0.9763103   0.55054242]\n",
            " [ 0.         -0.         -0.         -0.16929913  0.21209806]\n",
            " [-0.          0.          0.         -0.         -0.52165269]]\n",
            "[-0.9170418   1.40215838  1.41534372 -0.53305575 -1.02625922]\n"
          ]
        }
      ],
      "source": [
        "n = 5\n",
        "\n",
        "A = np.random.randn(n, n) * np.tri(n).T\n",
        "print(A)\n",
        "\n",
        "x = np.random.randn(n)\n",
        "print(x)\n",
        "\n",
        "b = A @ x"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "xcomp = np.zeros(n)\n",
        "\n",
        "for i in range(n-1, -1, -1):\n",
        "    tmp = b[i]\n",
        "    for j in range(n-1, i, -1):\n",
        "        tmp -= xcomp[j]*A[i,j]\n",
        "        \n",
        "    xcomp[i] = tmp/A[i,i]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now compare the computed $x$ against the reference solution."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[-0.9170418   1.40215838  1.41534372 -0.53305575 -1.02625922]\n",
            "[-0.9170418   1.40215838  1.41534372 -0.53305575 -1.02625922]\n"
          ]
        }
      ],
      "source": [
        "print(x)\n",
        "print(xcomp)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Questions/comments:\n",
        "\n",
        "* Can this fail?\n",
        "* What's the operation count?"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.5.1+"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}